Network RTK Quality Indication Using Linear Interpolation Residuals

Transcription

1 Network RTK Quality Indication Using Linear Interpolation Residuals September 2005 Published in proceedings of ION GNSS September 13-16, 2005, Long Beach, CA P. Alves, I. Geisler, N. Brown, J. Wirth, and H.-J. Euler

2 Introduction of a Geometry-Based Network RTK Quality Indicator P. Alves, I. Geisler, N. Brown, J. Wirth, and H.-J. Euler Leica Geosystems AG Heerbrugg, Switzerland BIOGRAPHIES Paul Alves graduated in 2005 with a Ph.D. in Geomatics Engineering form the University of Calgary, Canada. His research interests include network RTK, carrier phase processing, and ambiguity resolution. He is currently working as a software engineer for Leica Geosystems. Ines Geisler started studying Geodesy in 1999 at the Dresden University of Technology. She is currently doing her Diploma Thesis regarding the performance of Network RTK positioning dependent on different interpolation approaches. Aside of that, she is working as an Application Engineer in the Networked Reference Stations team of Leica Geosystems. Neil Brown graduated from the University of Melbourne in 1999 with a Bachelor of Arts and a Bachelor of Geomatics (Honours). In 2000 Neil commenced a Doctor of Philosophy degree at the Department of Geomatics, University of Melbourne in stochastic modelling of the GPS observables. Neil is currently working as a Product Manager in the Networked Reference Stations and Structural Monitoring team of Leica Geosystems whilst continuing to work on his PhD part-time. Joachim Wirth received his MSc in Mathematics from the University of Heidelberg and his PhD in Manufacturing technology from the Swiss Federal Institute of Technology in Zurich. After working in the field of medical image processing he joined Leica Geosystems in Hans-Juergen Euler studied Surveying at the Technical University of Darmstadt. He graduated in 1990 with a PhD on Fast GPS Integer Resolution in Small-Scale Networks. His research interests are in the field of Network RTK and application of the information in GNSS rovers. He serves in the standardization process of RTCM SC104 as chairman for Network RTK and Galileo. ABSTRACT The multiple reference station approach is widely known as a method for combining the data from a regional reference station network to provide precise measurement correction to users. This is performed by measuring the regional errors at the reference station locations and interpolating them for the location of the rover. The quality of those corrections is dependent on the reference station spacing, the location of the rover, and the characteristics of the measurement errors. This paper introduces a network RTK quality indicator based on the characteristics of the measurement errors. The indicator assumes that the more linear the regional correlated errors, the better the interpolation methods will perform. The linearity of the network measurement errors is measured and weighted based on the distance to the rover. This paper compares the proposed quality indicator to the performance of three interpolation methods: 2D surface, least squares collocation, and distance weighted interpolation. The interpolation methods provide an 18 to 69 percent improvement relative to the single reference station approach, however all the approaches perform and behave similarly. The quality indicator shown, successfully models the network RTK performance over time. The indicator can also produce reference station network quality maps, which are shown and analyzed. INTRODUCTION The quality of Network RTK corrections is a function of the following factors: network geometry, measurement errors, elimination of nuisance parameters (i.e. ambiguities), and the interpolation model that is used. All of these aspects are intermixed, for example, the interpolation model that is used should have the same spatial shape as the measurement errors. Alternatively, if the measurement errors are low then the reference stations can be located further away than if the measurement errors are high.

3 The factors that affect Network RTK performance are generally focused around the qualities and characteristics of the measurement errors. An accurate understanding of the measurement errors leads to an optimal interpolation model and network geometry for a given level of desired rover performance. The following literature focuses on measuring and quantifying the properties of the measurement errors (Alves, 2004; Wanninger, 2004; Wübbena, 2004; Chen, 2003; Raquet, 1998). In some of the more advanced cases the measurement error properties are extrapolated using network geometry to predict the performance for the rover in addition to the performance of the network. Many quality indicators for network RTK use the residual errors measured by the network reference stations to derive the current error conditions and characteristics (Chen, 2003; Wanninger, 2004). These characteristics are compared against the current interpolation model to determine the model residuals. For example, if the measured network residuals are linear and a linear interpolation model is used then there is a high likelihood that the rover will experience a high level of performance. The model residuals can then be used to predict the model inaccuracies as a function of the distance to the nearby reference stations. For example, if the model residuals are high but the rover is at a reference station then the model errors have no effect. The ability to determine the model residuals is a function of the degrees of freedom of the interpolation model. If there are no degrees of freedom then no residuals can be determined. In this case degrees of freedom can be created by excluding one of the reference stations from the model calculation. NETWORK RTK OVERVIEW Network-Based RTK methods use a network of reference stations to measure the correlated error over a region and to predict their effects spatially and temporally within the network. Although the name suggests that these methods are real-time specific, they can also be used in postmission analysis. This process can reduce the effects of the correlated errors much better than the single reference station approach, thus allowing for reference stations to be spaced much further apart thereby covering a larger service area than the traditional approach, while still maintaining the same level of rover performance. Network RTK is comprised of six main processes: 1. Processing of the reference station data to resolve the network ambiguities, 2. Selection of the reference stations that will contribute to the corrections for the rover, 3. Generation of the network corrections, 4. Interpolation of the corrections for the rover s location, 5. Formatting, and transmission of the corrections, and 6. Computation of the rover position. The main task of the network computation is to resolve the ambiguities for all stations in the network to a common ambiguity level, such that the bias caused by the ambiguities is cancelled when double differences are formed (Euler et al., 2001). The network correction computation uses the ambiguity leveled phase observations from the network reference stations to precisely estimate the differential correlated errors for the region. A subset of stations from the reference network, known as a cell, is selected to generate the correction for the rover based on the rover s position. One station in the cell, usually the one closest to the rover, is selected as the master station. The correction interpolation process models the network corrections to determine the effects of the correlated errors at the rover s position. Depending on the correction concept (master-auxiliary, VRS or FKP), the interpolation may be done either by the reference station software or the rover itself. The corrections are formatted in such a way that the rover or standard RTK software can interpret them. Note that until the release of the RTCM 3.0 network messages there has been no official standard for the transmission of network corrections. The interpolated corrections are then applied to the observations of the master station, which are then used, by the rover or RTK software to compute its position. METHODOLOGY The proposed method is a geometric approach to determining the linearity of the measurement errors. In single point positioning (no differencing) the absolute magnitudes of all measurement errors affect positioning and navigation performance. When differential methods are used, the difference in measurement errors and not the absolute values of the measurement errors dictate the level of performance. This brought forth the somewhat standard measure of parts per million of the baseline length for describing the magnitude of data set measurement errors.

4 In network RTK the linear trend of the errors is modeled and removed. Consequently, a large parts per million of the baseline length or error gradient value does not dictate the level of performance in network RTK if this gradient is consistent throughout the network. Figure 1 shows an example of this. The red line is the absolute measurement error, which have a high parts per million (indicated by its slope). The green line indicates the single reference station differential measurement error using the green station as the reference and the blue line shows the differential measurement error using the blue station as the reference. Assuming the network is using a linear interpolation model, the rover would have no measurement errors in network RTK mode. This is because the slope of the measurement errors is modeled in the network RTK interpolation model. Chen (2003) also briefly discusses this effect when discussing the ionosphere index I95 (Wanninger, 2004). Figure 2: Examples of the shape of a triangulated network. Top shows a network with flat measurement errors. Bottom shows a network with non-linear measurement errors. Absolute error Location Rover Reference stations Figure 1: Example of the difference between single reference station (shown in blue and green) and multiple reference station measurement error characteristics. The absolute measurement error is shown in red. The method described in this paper uses the change in the slope of the measurement errors as an indicator of the network RTK performance. This linearity is realized in two dimensions by triangulating the network and visualizing the network as a two dimensional surface or triangular facets. If the regional errors are linear then this surface will be flat (although it may not be horizontal) as shown in Figure 2 (top). Non-linearity in this surface appears as joints along the edges of the two dimensional surface (Figure 2, bottom). This method measures the change in the slope of the facets of a 2D triangulated network as an indication of the linearity of the measurement errors. PROCEDURE The calculation of this quality measure is performed in the following stages: 1. Triangulate the network. 2. Determine the triangle that contains the rover station and the corresponding three connecting stations. 3. Determine the three triangles that are connected to the edges of the rover s triangle. 4. Calculate the slopes of the surfaces for the triangle containing the rover and the three connected triangles using the network residuals for the connected stations. 5. Calculate the differences in the slopes along the direction of the intersection of the center surface with the outer surfaces. 6. Calculate the shortest distance between the rover position and the interface between the surfaces. 7. Weight the difference in slope values by the inverse of the distance to the surface boundary. Triangulation of the network can be performed through any of the available methods. In the cases shown, a Delaunay triangulation was used. The double difference slopes are calculated in the north and east directions using a plain fit to the double difference misclosures at the connecting reference stations. The single difference cannot be used because the station clocks will adversely affect the slopes. The difference in the slopes is calculated by using the dot product of a vector of the slopes. The vector

5 perpendicular to the surface is derived from the north and east slopes as follows n v = e (1) 1 where n is the slope in the north direction and e is the slope in the east direction. The difference in the slopes is calculated by s T v1 v2 12 = v1 v2 1. (2) The absolute value and minus one are added to change the range to a value between 0 and 2. This value represents the difference in the slopes and the linearity of the measurement errors across the local region of the network. The shortest distance between the rover and the three triangle borders is calculated to weight the three slope differences. The closer the rover is to an edge, the more weight this slope difference should have. The weighting used is the inverse of the distance to the nearest edge. The calculated values are all weighted together to give one value for each double difference as shown: s d + s d + s r12 13 r13 23 r 23 δ r = (3) d r12 + d r13 + d r 23 where δ r is the rover correction, d r, 12 is the shortest distance between the rover and the edge between slopes 1 and 2. This correction value is, in general, very small because it is a weighted cosine of the angle between the surfaces. To make these values practical for plotting they have all been multiplied by five hundred. The quality is a relative value that must be calibrated and then fixed. The relative values of the quality indicator are of importance. INTERPOLATION OF CORRECTIONS The quality values will be compared against a variety of interpolation methods to evaluate their effectiveness. This section describes the interpolation methods shown. There are a variety of methods of parameterization of the correction differences for the rover located within or near the network coverage area. All of the methods developed are based on function approximations. The methods used d in this investigation are a Distance-Based Linear Interpolation Method (Dai et al., 2004; Euler et al., 2004), a Low-Order 2D Surface Model (Dai et al., 2004; Euler et al., 2004; Wanninger, 2000; Fotopoulos & Cannon, 2001), and Least-Squares Collocation Method (Raquet, 1998; Marel, 1998; Alves, 2004). The aim of each of these approaches is to model the distance-dependent errors along the baseline between master and rover station on an epoch-by-epoch and satellite-by-satellite basis. As the input parameters for each interpolation algorithm are between-master and auxiliary single-differenced observables (either carrier-phase residuals for the L1/L2 frequencies or geometric-/ionospheric-free linear combinations), an n-1 independent error vector is generated where n is the number of reference stations involved. In addition to unambiguous (ambiguity leveled) correction differences, coordinates for all reference stations are also needed to either generate coordinate differences (when applying low-order surface models), or baseline lengths between the master and each further auxiliary station (when processing distance-weighted and least-squares collocation model). The vector of rover corrections is the result of a vector of reference station residuals (estimated errors) and a vector of coefficients (Dai et al., 2004) V rover V (4) where V rover is a vector of L1, L2, geometric-free or ionospheric-free frequencies, is a n-1 vector of coefficients and V is an n-1 vector of errors, Actual GPS measurements have to be observed in order to form the error vector. As a basic premise of all of the interpolation methods the coordinates of stations and satellites have to be known, and the ambiguities have to be correctly solved so that the correction differences share a common integer ambiguity level. All of the coefficients can be calculated from the known coordinates of the reference stations and are therefore derived from the geometry of the rover station and the network (Dai et al., 2004). Since the master-auxiliary concept is applied, the coefficient vector (as well as the error vector) refers to single baselines between a designated master and each auxiliary reference station. Once the correction differences are interpolated for the rover's location they are applied to the raw observations of the master reference station to improve the single baseline positioning between the master and rover station. Single baseline positioning based on network corrections in real-time mode is also referred to as Network RTK positioning.

6 The different interpolation methods and their characteristic determination of the coefficient vector are described below. Distance-Based Linear Interpolation Method The correction differences of master-auxiliary baselines are weighted by the inverse distance (one over the distance) of each auxiliary reference station to the rover. The further apart an auxiliary reference station is located from the rover station the less weight it gets. The equation i of a distance-based linear interpolation of CD rover can be defined as n 1 i CDk / S k i k= 1 CD rover = n 1 (5) 1/ S k= 1 k where n-1 is the number of auxiliary reference stations in the network, S k is the 3D coordinate distance between i reference station k and the rover, and CD k is the correction difference between auxiliary reference station k of satellite i and master station Equation (5) can be rewritten as where and w n i 1 k i CD rover = CDk (6) k= 1 w w k 1 = S k n 1 w k k= 1 (7) w = (8) From Equation (6) it follows that the coefficient vector α r is r w1 w2 wn α = L (9) w w w By definition, the values of the coefficient vector will always be between 0 and 1, regardless of the location of the rover with respect to the network. This interpolation method was originally proposed by Gao et al. (1997) to determine the ionospheric error at the rover's location. Dai et al. (2004) show that the distance-based interpolation can be applied to determine all distance-dependent errors, however only to a certain degree of accuracy. In comparison to other interpolation methods (low-order surface and least-squares collocation) the distance-based approach performs slightly worse because of estimating the errors in one dimension only. Low-Order Surface Model Low-order surface models are commonly used to determine distance-dependent errors by estimating a fitting function to generate an n-dimensional trend or regression surface. Because of a high degree of spatial correlation of distance-dependent errors 2D or 3D surfaces sufficiently model the major trend of the errors across the reference station network (Dai et al., 2004). The coefficients of the fitting functions can be determined for each coordinate dimension specifically and separately: easting, northing, and height. Thus the different error characteristics are taken into account. For instance, if a reference station network is characterized by large height differences it might be useful to model the tropospheric error by a 3D-fitting function because the impact of the tropospheric error is very much height dependent. In contrast, it may be sufficient to model the ionospheric error by applying a 2D-fitting function as shown V a X b Y c (10) The equations presented are only 1 st order fitting functions. Functions of a higher order (2 to n) can also be used. The coefficient vector is different depending on the number of variables (coordinate differences of latitude X, and longitude Y for example) and the number of orders. Usually, more than 3 reference stations are contributing to network processing, and therefore also to the determination of α r which requires a least-squares adjustment. If Equation (10) is used to model the trend surface, a minimum of 3 reference stations is required, and the coefficient vector is defined as (Dai et al., 2004): where X rover master Y rover master 1 A T A 1 A T (11)

7 A X 1 n Y 1 n 1 X 2 n Y 2 n 1 X n 1,n Y n 1,n 1 (12) Following the proposed Master-Auxiliary concept Equation (13) can be simplified by using single differences only. Thus the matrix B can be neglected assuming that the correction differences are uncorrelated. The simplified interpolation equation changes to After estimating α r, the distance dependent errors at the rover's location can be generated by applying the common formula shown in Equation (4). Least-Squares Collocation Method The least-squares collocation method models the distancedependent errors based on their spatial correlation. It is assumed that the further apart the reference stations are the less correlated the errors, which leads to a minimization of the errors in a least-squares sense. Leastsquares collocation is based on several assumptions (Alves, 2004): The generated correction differences are measurements of the true signal s biased by the noise n. The measurements (signal and noise) are zero mean with a normal distribution. The true signal s() t is unknown, but the variance-covariance of s( t) is known. The signal to be estimated is defined by its covariance, which can be described as the likelihood that two points will have the same values. The distance-dependent errors will be more similar with decreasing baseline length. Raquet (1998) proposes the following least-squares collocation equation for which double-differenced residuals between the master and auxiliary reference stations have to be calculated: s C sl s B T BC ll B T 1 A x Bl (13) where s is a vector of the estimated signals (corrections) at rover's position, C sl s is the covariance matrix between the signal components of the network observations and the predicted signal (interpolated vector s ), B is the observation matrix, which transforms C ll into double difference space, C ll is the variance-covariance matrix of the network observations (measurement vector l), A is the design matrix of the estimated parameters, x is the vector of estimated parameters, l is the vector of network observations. s C sl s C ll 1 l (14) By inverting C ll, low variance values become high and high variance values become low, and hence high weight is assigned to precise observations when 1 multiplying C ll by l. A covariance function is used to generate the covariance matrices. It calculates the estimated covariance between the observation's signal components. Depending on the characteristics of a covariance function, each function has a different shape and prediction characteristics. Alves (2004) shows that the greatest difference in the prediction characteristics between several covariance functions will be when the network is sparse or the correlation length is short. The following are the requirements that a covariance function should fulfill to best suit the error characteristics within a reference station network: The covariance function should produce a positive definite variance covariance matrix. The prediction should represent the likelihood that the errors measured at the reference stations are the same as the errors measured at the rover station. The covariance should converge to zero with increasing distance between stations. As a consequence, if no reference station is in a position to predict the distance-dependent errors at the rover's location, then the covariance of the signal and the network's observation signal should be zero. Exponential covariance functions have all of the characteristics (described above) for modeling the distance-dependent errors between the stations. In contrast to the proposed covariance function of Raquet (1998) and Marel (1998) which, for example, also takes the satellite's elevation into account, the function applied in this paper is a function of the distance between all reference stations only. There is no need to correct for the elevation of satellites because single differenced observations are introduced in the error modeling process, and the interpolation is done on a satellite-by-satellite basis. The exponential covariance function is

8 CF d ( d ) = e τ (15) where is the correlation length, and d is the 3Ddistance between all reference stations. Combining Equations (4) and (14) gives a coefficient vector of C sl s C ll 1 DATA SET DESCRIPTION (16) Data from selected reference stations in Lower Saxony, Germany was used for the evaluation of the interpolation methods and the quality indicator. This network is shown in Figure Figure 3: Reference station network in Lower Saxony, Germany. All distances shown in blue are in km. Data was processed with a one second data rate with an elevation mask of 13 degrees, however results will be shown in 30 seconds. This data set is from Oct. 31, 2003, which is the day after an ionospheric storm. The ionosphere for this short baseline reaches up to 2.5 parts per million of the baseline length. All the receivers in this network are Trimble 4700s. The rover station (0652) is located 19 km from the nearest reference station. The reference station baseline lengths range from 30 to 69 km. This is a medium scale network INTERPOLATION RESULTS The 2D model, least squares collocation, and distance weighted interpolation methods are compared against the single reference station approach for four double difference satellite pairs. These satellite pairs were selected because they show the greatest variation in terms of measurement errors for the data set. The distance-weighted model is the simplest model followed by the 2D model and the most complex model is the least squares interpolation method. However, the complexity of the model is not entirely correlated to the performance of the interpolation method. The RMS of the double difference L1 carrier phase residuals for the interpolation methods and the single reference station approach are shown in Table 1 along with the percentage of improvement. All of the methods perform well and show an improvement relative to the single reference station approach ranging from an 18 to 69 percentage of improvement, however the least squares model appears to provide the highest level of improvement followed by the distance weighted model. Dai et al. (2004) also show a similarity between the performance of various interpolation methods but they conclude that the distance weighted model performs slightly worse than the other methods. The results shown in Table 1 show that for this network under these error conditions that all the interpolation methods perform similarly, including the distance-weighted model. Table 1: RMS Error of the L1 double difference misclosures and the percentage of improvement for the single reference station and various interpolation methods. Single RS 2D Least Squares RMS Error (cm) Percent Improvement Distance Weighted Figures 4 to 7 show the double difference L1 phase residuals for the single reference station approach and the three interpolation methods. There is a high level of agreement between the interpolation methods. This shows that the interpolation methods all have the same behavior and the differences between them are small. In some cases the level of improvement is obvious in the figures. For example, satellite pairs 13 and 04 (Figure 5), and 08 and 13 (Figure 4) show an obvious improvement, while the improvement is less obvious in satellite pair 27 and 10 (Figure 6), for example. This is represented in the percentage of improvement shown in Table 1.

9 Figure 4: Double difference L1 carrier phase residuals for satellites 27 and 10 for the single reference station approach and the various interpolation methods. Figure 7: Double difference L1 carrier phase residuals for satellites 27 and 24 for the single reference station approach and the various interpolation methods. In general, the behavior and shape of the interpolation methods is similar. In the following examples only the 2D interpolation method will be shown as a representative of all the interpolation methods. QUALITY INDEX RESULTS Figure 5: Double difference L1 carrier phase residuals for satellites 13 and 4 for the single reference station approach and the various interpolation methods. The quality index values can be computed for every double difference satellite pair using only one epoch of data. The quality indicator is always positive and represents the potential magnitude of the network RTK measurement errors. A low value quality index means that the network errors are linear and that interpolation should be able to minimize the errors at the rover. A high value quality index means that the network errors are non-linear relative to the reference station scale and the interpolation may not be able to model the correlated errors. The double difference L1 carrier phase residuals of the single reference station and network RTK solutions are compared to the quality values in Figures 8 to 11. In general, there is a high level of correlation between the level of error in the network RTK solution and the quality value. This shows that this quality indicator is effective in monitoring network RTK performance. Figure 6: Double difference L1 carrier phase residuals for satellites 8 and 13 for the single reference station approach and the various interpolation methods. Figure 8 shows satellite pair 27 and 10. In this case the double difference residuals are higher at the beginning of the data set and decrease towards the end of the dataset. Once again, there is a high level of correlation between the quality value and the network RTK performance. The quality indicator follows the behavior of the network RTK performance.

10 more varied. This is shown as an increase in the quality indicator. Figure 8: Comparison of the quality value with the double difference residuals L1 carrier phase of the single reference station approach and the 2D interpolation for satellites 27 and 10. Figure 9 shows satellite pair 13 and 4. In this case the quality indicator closely matches the network RTK performance. At the beginning of the data set the network approach provides a high level of improvement over the single reference station approach. During this time the quality indicator shows a relatively low value, which matches the network RTK performance. This shows that the quality indicator is an indication of network RTK quality and not the quality of the single reference station RTK. Figure 10: Comparison of the quality value with the double difference L1 carrier phase residuals of the single reference station approach and the 2D interpolation for satellites 8 and 13. Figure 11 shows results for satellite pair 27 and 24. The beginning of the data set (from time to ) clearly shows that the quality indicator measures the network RTK performance and not the single reference station RTK performance. There is an event slightly after This error is shown in all of the results except for the quality value. This suggests that this error is localized to the rover station only. The magnitude of this error is small (approximately 3 cm) and is not consistent with a cycle slip. A low quality indicator shows that there is no network activity and, consequently the error is not reduced by the network corrections. The end of this data set shows an increase in the measurement errors that is followed closely by the quality value. Figure 9: Comparison of the quality value with the double difference L1 carrier phase residuals of the single reference station approach and the 2D interpolation for satellites 13 and 4. The next example, shown in Figure 10, is for satellite pair 8 and 13. In this case the quality indicator follows the network RTK performance less closely, however there is a noticeable change in behavior in the network RTK results, which is shown in the quality value. At the end of the data set the network RTK residuals become larger and Figure 11: Comparison of the quality value with the double difference L1 carrier phase residuals of the single reference station approach and the 2D interpolation for satellites 27 and 24.

11 MAP INDEX RESULTS In addition to a time series of the quality index, the quality can be displayed geographically. The following quality maps are produced by calculating the average quality value for all the satellites for one epoch for a grid of potential rover positions. Figure 12 shows the quality indicator map for one epoch. The fan pattern shown is due to the network geometry. In this case all of the triangles have only two adjacent triangles. If there is no connected triangle at an edge, then that edge is ignored. Consequently, the value changes only as a function of the distance from one edge to the other. This distance function is linear, which causes the linear gradient from one baseline to the next. The contour increase around the southern most station shows that this station is experiencing more error than the surrounding stations. This creates a ridge in the surface and, as a result, the high magnitude quality values around that station. The map in Figure 12 is interesting but unfortunately, not as useful for network RTK users because it shows the errors in the network and not the potential errors for a user. For example, a user standing beside the southern most station, with high quality values, would look at this plot and assume that they would have low network RTK performance. However, this user would experience a high level or performance because of their proximity to a reference station. The interpolation has very little effect when the rover is close to a reference station in the network. weight function has a value of one, which means that it has no effect on the quality value. Figure 13: The weighting function used to decrease the quality value when the rover moves closer the network reference stations. The map showing the expected network RTK performance as a function of rover location is shown in Figure 14. The influence of the southern most reference station is still shown, however the quality value for a rover that is close to the reference station will be small. The pockets of high quality values are for triangles with high quality values and locations furthest from the nearest reference station. Figure 12: Map of the quality index values for the network without reference station location weighting for a single epoch. To compensate for this effect a decreasing weight is added as a function of the rover to the surrounding reference stations. The weight function is shown in Figure 13. The weight function increases linearly for a roverreference station distance of 20 km. After 20 km the Figure 14: Map of the quality index values for the network with reference station location weighting for a single epoch. PROPOSED FUTURE WORK This paper shows that this quality indicator is effective in tracking and monitoring the quality of network RTK. One advantage of this method is that quality values can be calculated for each double difference independently, epoch-by-epoch. An interesting extension of this research would be to use to quality values in the position estimation filter. This could weight observations with a

12 low quality value higher than observations with a high quality value. Further analysis with a wider variety of measurement errors on multiple networks would help to validate the effectiveness of this index. CONCLUSIONS This paper proposes a geometry based quality index to monitor network RTK quality. This quality index can monitor the linearity of the instantaneous network errors in terms of each double difference measurement independently. Three interpolation methods were shown: 2D surface, least squares collocation, and distance weighted. These interpolation methods all show a moderate to high level of improvement ranging from 18 to 69 percent improvement relative to the single reference station approach. The measurement errors for the rover using all of the interpolation methods were similar in terms of magnitude and behavior. It can be concluded that for this data set and this network, there is very little difference between the three interpolation methods tested. The quality index was shown over time for four double difference satellite pairs and compared to the L1 carrier phase double difference residuals. In all the cases shown, the quality value closely follows the network RTK results. This shows that this geometry-based quality index is a good indicator for measurement quality after interpolating and applying the corrections. In addition, a map of the errors can be produced, which shows the expected network performance for a given rover position within the network. ACKNOWLEDGMENTS The authors would like to thank Volker Wegener of the Zentrale Stelle SAPOS, LGN, Hannover for kindly supplying us the data used for the numerical analysis. REFERENCES Alves, P. (2004) Development of Two Novel Carrier Phase-Based Methods for Multiple Reference Station Positioning, UCGE Report number 20203, Ph.D. Thesis, University of Calgary Chen, X., H. Landau, U. Volloth (2003) New Tool for Network RTK Integrity Monitoring, Proceedings of the National Technical Meeting of the Satellite Division of the Institute of Navigation ION GPS 2003 (September 2003, Portland, USA), Dai, L., Han, S., Wang, J., Rizos, C. (2004) Comparison of Interpolation Techniques in Network-Based GPS Techniques, Journal of The Institute of Navigation, Vol. 50, No. 4, Euler, H.-J., C.R. Keenan, B.E. Zebhauser, G. Wübbena, (2001) Study of a Simplified Approach in Utilizing Information from Permanent Reference Station Arrays, Proceedings of National Technical Meeting of the Institute of Naviagation, ION GPS 2001 (September 2001, Salt Lake City, USA), Euler, H.-J., S. Seeger, O. Zelzer, F. Takac, B.E. Zebhauser (2004) Improvement of Positioning Performance Using Standardized Network RTK Messages, Proceedings of the Institute of Navigation National Technical Meeting, ION NTM 2004 (January 2004, San Diego, USA), Fotopoulos, G. and Cannon, M. E. (2001) An Overview of Multi-Reference Station Methods for Cm-Level Positioning, GPS Solutions, Vol. 4, No. 3, 1-10, 2001 Gao, Y., Z. Li and J.F. McLellan (1997) Carrier Phase Based Regional Area Differenctial GPS for Decimetre-Level Positioning and Navigation, Proceedings of the National Technical Meeting of the Satellite Division of the Institute of Navigation, ION GPS 1997 (September 1997, Kansas City, USA), Marel, H. van der (1998) Virtual GPS Reference Stations in the Netherlands, Proceedings of the National Technical Meeting of the Satellite Division of the Institute of Navigation, ION GPS 1998 (September 1998, Nashville, USA), Raquet, J. (1998) Development of a Method for Kinematic GPS Carrier-Pahse Ambiguity Resolution Using Multiple Reference Receivers. UCGE Reports 20116, Ph.D. Thesis, University of Calgary, Canada. Wanninger, L. (2000) Präzise Positionierung in regionalen GPS-Referenzstationsnetzen, PhD Thesis, published by Deutsche Geodätische Kommission, series C, booklet 508, Munich 2000, Germany

13 Wanninger, L. (2004) Ionospheric Disturbance Indicies for RTK and Network RTK Positioning, Proceedings of the National Technical Meeting of the Satellite Division of the Institute of Navigation, ION GPS 2004 (September 2004, Long Beach, USA), Wübbena, G., Schmitz, M., Bagge, A. (2004) GNSMART Irregularity Readings for Distance Dependent Errors, Geo++ White Paper, Garbsen, Germany.

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